After creating two line segments that are touching in the shape of a V, is there an extension to draw an angle in degrees inside of that vector? So, inside of that vector there will be an arc with the number of the degrees and the degree symbol displayed.

There's a tool that will display that info, but it doesn't draw the arc or print the numbers/text. That's the new Measurement tool (new in version 0.91). And there's an extension that will do exactly what you want, except only for length and area measurements - not for angle.

It might not be very hard to add angles to that extension (which is Extensions menu > Visualize Path > Measure Path. I don't know how to do that, but for I mean for people who do, it might not be very hard.

There might be an extension out there somewhere, that I haven't found yet. I guess you've already searched the internet?

Well, lacking any extension, it still wouldn't be very hard to do this. First, use the Measurement tool to acquire the angle. Then use the Ellipse tool, in Arc Mode, to draw the arc. And finally, the Text tool, to type the angle.